This statement is contained in [1]}.
We briefly explain a proof. We can assume that \(\mathcal {F}\)  and \(\mathcal {E}\)  are locally free sheaves.
The vanishing (REF ) is then clear for \(a\ne 0\) .
Let \(\mathcal {Z}=R^{\lambda \geqslant 0}/\mathbb {C}^*\) .
For \(a=0\) , it suffices to check the statement for locally free sheaves on \(\mathcal {Z}\) .
We can assume that \(\mathcal {F}=\mathcal {O}_{\mathcal {Z}}(u)\)  and \(\mathcal {E}=\mathcal {O}_{\mathcal {Z}}(v)\)  with \(v\leqslant w<u\) .
We have that:
\(\text{Hom}_{\mathcal {Z}}\left(\mathcal {O}_{\mathcal {Z}}(u), \mathcal {O}_{\mathcal {Z}}(v)\right)\cong \left(\mathbb {C}\left[\left(R^{\lambda \geqslant 0}\right)^{\vee }\right](v-u)\right)^{\mathbb {C}^*}=0.\) 
